If `y = 2^(x) " then " (dy)/(dx) =`?

Options

A. `x(2^(x -1))`
B. `(2^(x))/((log 2))`
C. `2^(x) (log 2)`
D. none of these

Explanation

To find the derivative of the function \( y = 2^x \), we can follow these steps:

### Step 1: Take the logarithm of both sides
We start with the equation:
\[ y = 2^x \]
Taking the natural logarithm on both sides gives us:
\[ \log y = \log(2^x) \]

### Step 2: Use the properties of logarithms
Using the property of logarithms that states \( \log(a^b) = b \log a \), we can rewrite the right-hand side:
\[ \log y = x \log 2 \]

### Step 3: Differentiate both sides
Now, we differentiate both sides with respect to \( x \). Remember that when differentiating \( \log y \), we need to use implicit differentiation:
\[ \frac{d}{dx}(\log y) = \frac{1}{y} \frac{dy}{dx} \]
On the right side, since \( \log 2 \) is a constant, the differentiation gives:
\[ \frac{d}{dx}(x \log 2) = \log 2 \]

### Step 4: Set the derivatives equal
Now we can set the derivatives equal to each other:
\[ \frac{1}{y} \frac{dy}{dx} = \log 2 \]

### Step 5: Solve for \( \frac{dy}{dx} \)
To isolate \( \frac{dy}{dx} \), we multiply both sides by \( y \):
\[ \frac{dy}{dx} = y \log 2 \]

### Step 6: Substitute back for \( y \)
Since we know that \( y = 2^x \), we can substitute back:
\[ \frac{dy}{dx} = 2^x \log 2 \]

Thus, the final result is:
\[ \frac{dy}{dx} = 2^x \log 2 \]

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